While most universities boast no more than 50,000 students, Keith Devlin's class--Introduction to Mathematical Thinking--has almost 60,000 students enrolled. Even if only one-tenth of the students complete the course, it's still a massive number of people. This huge enrollment is made possible by the format of the MOOC (Massively Open Online Course). There are 11 math MOOCS currently being offered on Coursera (the site hosting Devlin's course). These courses are free, create an opportunity for people to learn from professors at prestigious universities, and are becoming interesting to universities, especially for the purpose of teaching remedial courses. Managing such a massive course seem mind-boggling, but Devlin has deputized 900 "Teaching Assistants" and created discussion boards to facilitate students' learning. Devlin chronicles his experiences, including his "mistakes," on his blog mooctalk.

Whether you are inspecting your fingerprints, your roster of friendships, or the proportions of your body, Steven Strogatz wants you to know that mathematical patterns are key to explaining your observations. In his series, "Me, Myself, and Math," he has so far explored the topology of the fingerprint using index theory, the idea that your friends have more friends than you do, and most recently, the "divine" proportion phi (the golden ratio), which is purported to be the ratio of a man's height to the distance from the floor to his navel. After divulging his own measurements as being non-golden, Strogatz debunks the ubiquitousness of phi while still highlighting its geometric beauty. Prompting over a hundred comments, his blog entry seems to have caught the attention of many readers (even if many of them wrote in only to disagree with his categorization of phi as the second most popular number). In addition to being well-written, each piece includes an annotated list of references (both online and hardcopy) for those whose curiosity was roused. See all posts by Strogatz.

This charming piece describes a very new result on that old chestnut, the Prisoner's Dilemma, a two-player game dating to 1950 which is a staple of game theory, and has long been used to study cooperation in an evolutionary context. The new result comes to us from the computational biologist William H. Press--the president of the American Association for the Advancement of Science, as well as a professor at the University of Texas at Austin--and the physicist Freeman Dyson--a scientist with a wide-ranging intellect, most famous for his work on quantum electrodynamics. The pair of former wunderkind, as Bartlett amusingly describes them, published what appears to have been a weekend's worth of work in May's Proceedings of the National Academy of Sciences, and since then their paper, "Iterated Prisoner’s Dilemma contains strategies that dominate any evolutionary opponent," has been met with a multitude of proclamations about the demise of cooperation, which this piece falls a little short of debunking.

While the Prisoner's Dilemma was originally proposed by mathematicians Merrill Flood and Melvin Dresher of the RAND corporation in 1950, its name comes from mathematician Albert Tucker's formulation involving two criminals. These criminals are arrested, but the police do not have enough information to convict them for their crime. As a result, if neither man confesses, both will be sentenced to a month in jail on a minor charge. If one man testifies against his partner, while the partner remains silent (and so "cooperates"), the betrayer will be granted immunity and go free while his fellow perpetrator gets a one-year sentence. Finally, if both men testify, each will receive a three-month sentence. If we assume each man wants to minimize his own jail time, his only option is to rat out his colleague; whether his partner betrays him or cooperates with him, he will receive less jail time if he himself defects. The inevitable outcome is that the two criminals rat on each other, and get three months in jail--more jail time than if they had cooperated--and that rational self-interest yields worse outcomes for both players than altruism (something only an economist could find surprising).

The link between the Prisoner's Dilemma and the evolution of cooperation was made by Robert Axelrod in his 1984 book, called, ahem, The Evolution of Cooperation. Axelrod organized a tournament in which players competed at the iterated Prisoner's Dilemma--a more realistic game in which two players play a large number of rounds of Prisoner's Dilemma, and can base their decision to cooperate or defect on their opponent's history. The contestants in the tournament were not criminals, or even mathematicians, but computer programs. The one that won (or is it the other way around?) was designed by mathematician Anatol Rapoport. Called Tit-for-Tat, this simple program starts out cooperating, and then copies its opponent's previous move--rewarding cooperation and punishing defection. Axelrod went on to provide rigorous proofs of Tit-for-Tat's winning ways. For example, a population of Tit-for-Tatters is stable against single invaders, and groups of invaders, with a different strategy. Other strategies, such as the self-explanatory Always Defect, are stable against single invaders, but not groups. And a group of Tit-for-Tatters, cooperating with each other, can invade a population of Always Defectors--leading to the dominance of altruistic cooperation over rational self-interest in this evolutionary context.

Press and Dyson's result introduces a new family of strategies for iterated Prisoner's Dilemma which are capable of beating every other strategy--including both cooperative and greedy strategies. Using some elementary linear algebra, the two show that there exists a strategy, followed by one player, which either fixes a linear combination of the long-run scores of the players, or the ratio of the players' scores. As a result, strategies can be derived, using simple algebra, to fix your opponent's average score, or to fix the ratio between your score and your opponent's. Thus, these strategies--called Zero Determinant strategies--will win against any opponent in long enough games. The rub, as explained in an excellent blog and a new paper on Zero Determinant strategies, is that these strategies don't win in the same way Tit-for-Tat wins. Populations of Zero Determinators (or maybe Zero-d Terminators?) can be invaded by other strategies, and in the long run, are bound to evolve into less coercive strategies. And in tournaments, winning Zero Determinant strategies do so at the cost of earning very few points--just like Always Defectors. In other words, the tricks you should pick depend on the kinds of spoils you're after. If you prize winning the competition above all else, you might go for a Zero-d strategy. If you'd rather have a smaller piece of a bigger pie, though, you might want to pick a more cooperative strategy. [Note: The online version of this piece is titled, "To the Trickster Go the Spoils."]

In this retrospective, Stanford University professor of psychology James McClelland describes the pioneering career of recently deceased Robert Duncan Luce, "a mathematician who sought to provide axiomatic formulations for the social sciences." After earning his PhD in mathematics from MIT in 1950, Luce's "first publication applied ideas from [abstract algebra] to provide a mathematical definition of a 'clique' within a social network, and explored using a matrix to capture connections among individuals that might also be represented in a group," a method that "has become standard in computer science." At the same time, Luce addressed the problem of choice: in his 1959 book Individual Choice Behavior: A Theoretical Analysis, Luce "provided the foundation for a vast range of investigations in psychology and economics." From the 1960s to the present, "Luce's focus shifted to fundamental questions of measurement, with a particular emphasis on measuring psychological quantities such as value or loudness...Luce sought to establish fundamental principles relevant to the measurement of these psychological quantities."

Luce's "many substantive, institutional, and person contributions to the mathematical social sciences...were recognized early in his career," notes McClelland, "with election to the National Academy of Sciences in 1972, and later at the highest level, with the U.S. National Medal of Science in 2003."

Networks are complicated enough but mathematicians and others are now investigating networks of networks to understand systems of interconnected systems ranging from cells to the Internet. Quill talked to several researchers in different fields and gives many examples of networks that are connected to one another. Some researchers are interested in what causes systems to have catastrophic failure, such as what happened to the world economy in 2008, while others study the dynamics that keep networks, such as power grids, functioning. Alessandro Vespignani, a physicist and computational scientist at Northeastern University, says that it's a new field for which "We need to define new mathematical tools...We need to gather a lot of data. We need to do the exploratory work to really chart the territory." [Hear Jon Kleinberg (Cornell University) talk about his research in network theory in the Mathematical Moment podcast "Finding Friends."]

The Science Friday Book Club meets to discuss Edwin Abbott's classic novel published in 1884, Flatland: A Romance of Many Dimensions. Ian Stewart, emeritus professor of mathematics at the University of Warrick in England, joins host Ira Flatow, multimedia editor Flora Lichtman and senior producer Annette Heist. Ian Stewart is also the author of the book Flatterland: Like Flatland, Only More So and The Annotated Flatland: A Romance of Many Dimensions. Flatland plays around with the concept of dimensions and tells the story of a two-dimensional world where women are straight lines and men are polygons. The creatures of this two-dimensional world are faced with the daunting task of trying to comprehend the third dimension. Ian Stewart sums up the message of the book by saying "I think it is the message of realizing that your own particular parochial little bit of universe is not necessarily everything there is, and that you really should keep an open mind, but not so open that your brains fall out."

In an era when the importance of extremist viewpoints is increasing, Steve Strogatz created a model of zealots and moderates to study how the population of each group changes over time. Strogatz’s initial model was based on encounters between a "listener" and a "speaker," chosen at random from a group of moderates or two different viewpoint camps, with the condition that a speaker from one viewpoint camp would bring a moderate listener over to his own side or turn an opposing viewpoint listener into a moderate, while a moderate speaker would have no effect. The model demonstrated that if one group of zealots constitutes a sufficiently small percentage of the overall population, then the reigning viewpoint will prevail; when the percentage of zealots reaches a certain threshold, however, almost the entire population is converted to the zealot viewpoint. In an effort to increase the size of the moderate camp, Strogatz and his collaborators tried a variety of scenarios and found that only allowing moderates to evangelize was successful. The research is in the paper "Encouraging moderation: Clues from a simple model of ideological conflict," Seth A. Marvel, et al.

In this article, writer Kerstin Nordstrom describes some of the work of Laura Miller, a University of North Carolina at Chapel Hill professor of mathematics and biology "who studies the flow of fluids in (or around) living creatures. One of Miller's projects looks at blood flow in the embryonic heart and lays the groundwork for people to surgically correct heart defects, possibly in utero." Blood flow in an embryo's heart tube appears to directly influence the development of heart chambers and heart valves in the heart. "It is…thought that many congenital heart diseases may begin to appear at this critical state," Miller notes. Her group has studied this problem by taking measurements in actual animals, as well as using mathematical and physical models. "In her lab's simulations, they can easily tweak different conditions and see what effect it has on the flow rate and pattern. They can verify their simulations by looking at experimental systems—not actual hearts, but Plexiglas models that capture the same physics," Nordstrom writes.

Is it a bird? Is it a plane? No it is the mathematical constant pi (3.14159..) in the sky. On September 12th five aircrafts used dot-matrix skywriting technology to write out a thousand digits of pi stretching for a 100-mile loop each with numeral standing a quarter-mile tall. This endeavor was the brainchild of California artist Ishky as a part of the 2012 Zero1 Biennial, an event that celebrates the convergence of contemporary art and technology. On a Facebook page devoted to the Pi in the Sky affair, the event is said to "explore the boundaries of scale, public space, permanence, and the relationship between Earth and the physical universe. The fleeting and perpetually incomplete vision of pi's never-ending random string unwinding in the sky will create a gentle provocation to the Bay Area's seven million inhabitants."

In 2005, University of California San Diego physics professor Jorge Hirsch invented an index, generally known as the h-index, for quantifying a scientist's publication productivity. Noting that the h-index and similar indices only capture "past accomplishments, not future achievements," the authors of this article describe their work developing a formula to predict a scientist’s future h-index, based on the information available in a typical CV. They began with a large initial sampling of neuroscientists, Drosophila researchers, and evolutionary scientists. The application of several restrictions to this group reduced the sampling to "3,085 neuroscientists, 57 Drosophila researchers and 151 evolutionary scientists for whom we constructed a history of publication, citation and funding." Then, "for each year since the first article published by a given scientist, we used the features that were available at the time to forecast their h-index." These features included the number of articles written, the current h-index, the number of years since publishing the first article, the number of distinct journals published in, and the number of articles in prestigious neuroscience journals. The resulting formulas, for neuroscientists in particular, yielded "respectable" predictions, and showed that while the importance of the current h-index in predicting future h-indices decreased over time, "the number of articles written, the diversity of publications in distinct journals and the number of articles published in five prestigious journals all became increasingly influential over time."

On August 30th, Shinichi Mochizuki uploaded a series of four mathematics papers totaling about 500 pages to his website, and catapulted himself to fame. With the final paper, Mochizuki--a highly-acclaimed number theorist from Kyoto University in Japan--claimed to have proven the abc conjecture, which Princeton University number theorist Dorian Goldfeld of Princeton University has called "the most important unsolved problem in Diophantine analysis". The abc is a kind of grand unified theory of Diophantine curves: "The remarkable thing about the abc conjecture is that it provides a way of reformulating an infinite number of Diophantine problems," says Goldfeld, "and, if it is true, of solving them." Proposed independently in the mid-80s by David Masser of the University of Basel and Joseph Oesterle of Marie Curie University, the abc conjecture describes a kind of balance or tension between addition and multiplication, formalizing the observation that when two numbers a and b are divisible by large powers of small primes, a + b tends to be divisible by small powers of large primes. The abc implies--in a few lines--the proofs of many difficult theorems and outstanding conjectures in Diophantine equations-- including Fermat's Last Theorem. "No wonder mathematicians are striving so hard to prove it," said Goldfeld in 1996, "like rock climbers at the base of a sheer cliff, exploring line after line of minute cracks in the rock face..." The possibility of finally reaching this summit has led to reports in countless media outlets, and serious attention from luminaries like Terence Tao and Minhyong Kim. But Mochizuki has done much more than prove the abc conjecture--instead of toiling alongside his fellow climbers, Mochizuki seems to have built an airplane to take him to the top of the abc cliff and far beyond. Mochizuki's papers (bottom of the page) are the fruit of at least ten years of intense focus. They develop a theory Mochizuki calls "inter-universal Teichmuller theory." "He’s taking what we know about numbers and addition and multiplication and really taking them apart," says Kim in the Times. "He creates a whole new language--you could say a whole new universe of mathematical objects--in order to say something about the usual universe." Perhaps the work is more like a UFO than an airplane: mathematician Jordan Ellenburg of the University of Wisconsin, Madison, has described reading Mochizuki's magnum opus as "a bit like ... reading a paper from the future, or from outer space." The abstracts of the first and fourth paper are worth reading, for gems like this one from the close of the last abstract: "Finally, we examine the foundational/set-theoretic issues surrounding [the new theory] by introducing and studying the basic properties of the notion of a 'species,' which may be thought of as a sort of formalization, via set-theoretic formulas, of the intuitive notion of a 'type of mathematical object'."

The four papers, which heavily reference Mochizuki's own prior work developing theories including "p-adic Teichmuller theory," "Abstract anabelian geometry," and "The Hodge-Arakelov theory of elliptic curves," are out of the realm of understanding of even expert number theorists. "Most of the people who say positive things about it cannot say what are the ingredients of the proof," says Nets Katz of Indiana University. But while it's too soon to draw any conclusions--and may be for several years--there is reason to be optimistic, as Mochizuki "has a long track record," says Ellenberg, "and he has a long track record of being original." University of Tokyo professor Yujiro Kawamata concurs, calling Mochizuki a "researcher who has built unique theories on his own and uses singular terminologies in his often voluminous papers."

Traditional statistics has a measure, a "p value," for the likelihood that you will observe a given outcome (for example, obtaining 4 heads in 10 coin tosses, for example), but there is no absolute scale for measuring how well observed "evidence" (tossing 4 heads) correlates with a hypothesis (that the coin is unfair). This is where statistical geneticist Veronica Vieland’s work comes in. Starting with the Kelvin temperature scale as a base, Vieland drew parallels between molecules in a gas, which are responsible for temperature, and units of data from repeated trials, which are the "evidence" to be weighed, in order to create a so-called equation of state for evidence. Vieland’s equation is designed such that the addition of new information has a consistent effect on the overall strength of the evidence, and it allows for objective determination of how strongly a set of evidence supports different hypotheses. Vieland notes that her work is only a first step in developing an absolute measure of evidence strength, but it shows that such a measure can be found.

This short interview with Glen Whitney manages to be a sweet gloss on New York's soon-to-be-opened Museum of Math (MoMath), which will become the only North American museum dedicated exclusively to mathematics on December 15th. Whitney, MoMath's brainfather, describes his own career in mathematics by saying "I had a voracious appetite for math in high school but ... no illusion that I was going to be one of the top researchers in the country." Instead, he found a niche working in statistical trading at Renaissance Technologies. He left that job four years ago wanting to do something with a broader impact. Whitney hopes that by filling the vacuum in modern mathematical content at existing museums--one of the rare existing exhibits about math has inhabited the New York Hall of Science since 1960--MoMath will combat a widespread prejudice against mathematics. But MoMath will also be different from other mathematics outreach efforts, having a focus on physical interaction and "whole-body involvement," a broad perspective on mathematics, and striving to produce in visitors the thrill of discovery--the "Aha!" moments that make mathematics so exciting. These factors also distinguish Whitney's mathematical walking tours, in which he talks about the geometry underlying architecture and natural patterns, the mathematics of the subway and the algorithms that control traffic lights. These walking tours are as open-ended as mathematics itself: says Whitney, "If you give me a route, I'll make a tour. There is maths everywhere." (Photo: Rendering of museum's upper level, courtesy of the Museum of Mathematics.)

German artist Albrecht Dürer’s contributions to the world include not only famous prints, but also a famous book for both artists and mathematicians: The Painter’s Manual. With detailed yet accessible descriptions of geometric concepts first written by Euclid and Ptolemy, and elaborations on the construction of parabolas and ellipses, Dürer’s book represented his own study of the math behind representations of the world, undertaken in progressive scientific times. This article follows Dürer’s life from his humble beginnings as a son of a goldsmith, through his friendship with a wealthy purchasing agent who provided access to rare books, his fascination with a geometrically based drawing of male and female figures, and his assessment that German artists could rise to the level of their Italian peers only through study of geometric principles. The article also provides an overview of the historical development of conical sections, the creation of ellipses, parabolas, and hyperbolas by intersecting a plane with a cone. Image from the Albrecht Dürer Wikipedia page.